Read RD Sharma Solutions Class 9 Chapter 20 Surface Area and Volume of A Right Circular Cone below, students should study RD Sharma class 9 Mathematics available on Studiestoday.com with solved questions and answers. These chapter wise answers for class 9 Mathematics have been prepared by teacher of Grade 9. These RD Sharma class 9 Solutions have been designed as per the latest NCERT syllabus for class 9 and if practiced thoroughly can help you to score good marks in standard 9 Mathematics class tests and examinations

**Exercise 20.1**

**Question 1: Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.**

**Solution:**

Slant height of cone (l) = 60 cm

Radius of the base of the cone (r) = 21 cm

CSA of the right circular cone = πrl

CSA of the right circular cone = 22/7 × 21 × 60

CSA of the right circular cone = 3960 cm

^{2}Hence, the CSA of the right circular cone is 3960 cm

^{2}**Question 2: The radius of a cone is 5cm and vertical height is 12cm. Find the area of the curved surface.**

**Solution:**

Radius of cone (r) = 5 cm

Height of cone (h) = 12 cm

l

^{2}= r^{2}+ h^{2}l

^{2}= (5)^{2}+(12)^{2}l

^{2}= 25 + 144l

^{2}= 169l = 13 cm

CSA of a cone = πrl

CSA of a cone = 3.14 × 5 × 13

CSA of a cone = 204.28

Hence, the CSA of the cone is 204.28 cm

^{2}**Question 3 : The radius of a cone is 7 cm and area of curved surface is 176 cm**

^{2}. Find the slant height.**Solution:**

It is given that,

Radius of cone(r) = 7 cm

Curved surface area = 176cm

^{2}CSA of a cone = πrl

176 = πrl

176 = 22/7×7×l

(176×7)/(22×7) =l

8=l

l=8

Hence, the slant height of the cone is 8cm.

**Question 4: The height of a cone 21 cm. Find the area of the base if the slant height is 28 cm.**

**Solution:**

It is given that,

Height of cone (h) = 21 cm

Slant height of cone (l) = 28 cm

r

^{2}=l^{2}-h^{2}r

^{2}= (28)^{2}+ (21)^{2 }r

^{2}= 282 – 212r= 7√7 cm

Area of the circular base = πr

^{2}Area of the circular base = 22/7 × (7√7 )

^{2}Area of the circular base = 22/7 × 343

Area of the circular base = 1078cm

^{2}Hence, the area of the base is 1078cm

^{2}.**Question 5: Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.**

**Solution:**

It is given that,

Radius of cone (r) = 6 cm

Height of cone (h) = 8 cm

l

^{2}= r^{2}+ h^{2}l

^{2}= 6^{2}+8^{2}l

^{2}= 36 + 64l

^{2}= 100l = 10cm

TSA of the cone = CSA of cone + Area of circular base

TSA of the cone = πrl + πr

^{2}TSA of the cone = (22/7×6×10)+ (22/7×6×6)

TSA of the cone = 1320/7 + 729/7

TSA of the cone = 301.71cm

^{2}Hence, the area of the base is 301.71cm

^{2}.**Question 6: Find the curved surface area of a cone with base radius 5.25 cm and slant height 10 cm.**

**Solution:**

Radius of the cone (r) = 5.25 cm

Slant height of the cone (l) = 10 cm

Curved surface area (CSA) = πrl

Curved surface area = 22/7 × 5.25 × 10

Curved surface area = 165

Hence, curved surface area of the cone is 165cm

^{2}.**Question 7: Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.**

**Solution:**

Diameter of the cone (d) = 24m

Radius of the cone(r) = diameter/2 = 224/2 m = 12m

Slant height of the cone (l) = 21m

TSA of cone = CSA of cone + Area of circular base

TSA of cone = πrl+ πr

^{2}TSA of cone = (22/7×12×21) + (22/7×12×12)

TSA of cone = (22/7×252) + (22/7×144)

TSA of cone = (5544/7) + (3168/7)

TSA of cone = 792 + 453.57

TSA of cone = 1244.57

Hence, TSA of the cone is 1244.57m

^{2}.**Question 8: The area of the curved surface of a cone is 60 π cm**

^{2}. If the slant height of the cone be 8 cm, find the radius of the base.**Solution:**

It is given that,

Curved surface area = 60π cm

^{2}Slant height of the cone (l) = 8 cm

Curved surface area = πrl

πrl = 60π

r × 8 = 60

r = 60/8

r = 7.5

Hence, radius of the base of the cone is 7.5 cm.

**Question 9: The curved surface area of a cone is 4070 cm**

^{2}and diameter is 70 cm .What is its slant height?**(Use π= 22/7)**

**Solution:**

It is given that,

Diameter of the cone (d) = 70 cm

Radius of the cone (r) = diameter/2 = 70/2 cm = 35 cm

Curved surface area = 4070 cm

^{2}We know that, Curved surface area = πrl

Curved surface area = 4070

πrl = 4070

22/7×35×l=4070

22×5×l=4070

110×l=4070

l=4070/110

l=37

Hence, the slant height of cone is 37cm.

**Question 10: The radius and slant height of a cone are in the ratio 4:7. If its curved surface area is 792cm**

^{2}, find its radius. (Use π =22/7)**Solution:**

It is given that,

CSA of cone = 792 cm

^{2}Ratio of Radius and slant height of a cone is 4:7

Let 4x be the radius and 7x be the height of cone.

CSA of cone = πrl

22/7 × (4x) × (7x) = 792

22/7 × 28x

^{2}= 79222 × 4x

^{2}= 792x

^{2}= 9x = 3

Radius = 4x

Radius = 4(3) cm

Radius = 12cm

Hence, the radius of cone is 12cm

**Exercise 20.2**

**Question 1: Find the volume of the right circular cone with:**

**(i) Radius 6cm, height 7cm**

**(ii) Radius 3.5cm, height 12cm**

**(iii) Height is 21cm and slant height 28cm**

**Solution:**

(i) Radius of cone (r) = 6cm

Height of cone (h) = 7cm

Volume of a right circular cone = 1/(3πr

^{2}h)Volume of a right circular cone = 1/3 x 3.14 x 6

^{2}x 7Volume of a right circular cone = 264

Volume of a right circular cone is 264 cm

^{3}(ii) Radius of cone (r) = 3.5 cm

Height of cone (h) = 12cm

Volume of a right circular cone = 1/3πr

^{2}hBy substituting the values, we get

Volume of a right circular cone = 1/3 x 3.14 x 3.5

^{2}x 12Volume of a right circular cone = 154

Volume of a right circular cone is 154 cm

^{3}(iii) Height of cone (h) = 21 cm

Slant height of cone (l) = 28 cm

r

^{2}= l^{2}- h^{2}r

^{2}= (28)^{2}– (21)^{2}r = 784 – 441

r = 343

r = 7√7

Volume of a right circular cone = 1/3 πr

^{2}hVolume of a right circular cone = 1/3 x 3.14 x (7√7)

^{2}x 21Volume of a right circular cone = 7546

Hence, the volume of a right circular cone is 7546 cm

^{3}**Question 2: Find the capacity in litres of a conical vessel with:**

**(i) radius 7 cm, slant height 25 cm**

**(ii) height 12 cm, slant height 13 cm.**

**Solution:**

(i) Radius of the cone(r) =7 cm

Slant height of the cone (l) =25 cm

l

^{2}= r^{2}+ h^{2}25

^{2}= 7^{2}+ h^{2}25

^{2}- 72 = h^{2}625 – 49 = h

^{2}576 = h

^{2}√576 = h

h = 24

Volume of a right circular cone = 1/3πr

^{2}hVolume of a right circular cone = 1/3 × 22/7 × (7)

^{2}× 24Volume of a right circular cone = 1232

Volume of a right circular cone is 1232 cm

^{3}.Hence, the value of a right circular cone is 1.232 litres.

(ii) Height of cone (h) = 12 cm

Slant height of cone (l) = 13 cm

l

^{2}= r^{2}+ h^{2}13

^{2}= r^{2}+ 12^{2}13

^{2}- 12^{2}= r^{2 }169 – 144 = r

^{2}25 = r

^{2}r = 5

Volume of a right circular cone = 1/3πr

^{2}hVolume of a right circular cone = 1/3 × 22/7 × (5)

^{2}× 12Volume of a right circular cone = 314.28

Volume of a right circular cone is 314.28 cm

^{3}.Hence, the value of a right circular cone is 0.314 litres.

**Question 3: Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes.**

**Solution:**

Height of first cones = h

Height of second cones = 3h

Radius of first cones = 3r

Radius of second cones = r

**Question 4: The radius and the height of a right circular cone are in the ratio 5 : 12. If its volume is 314 cubic meter, find the slant height and the radius. (Use π=3.14).**

**Solution:**

Let us assume the ratio of radius and the height of a right circular cone to be x.

Then, radius be 5x and height be 12x

l

^{2}= r^{2}+ h^{2}l

^{2}= (5x)^{2}+ (12x)^{2}l

^{2}= 25x^{2}+ 144x^{2}l

^{2}= 169x^{2}l = 13x

Volume of cone = 314 m

^{3}1/3 πr

^{2}h = 3141/3 × 3.14 × (25x

^{2}) × (12x) = 31425x

^{2}× 12x = 314 × 3 × 3.14300x

^{3}= 314 × 3 × 3.14x

^{3}=1x = 1

Radius = 5 × 1

Radius = 5 m

Hence, the slant height and radius is 13m and 5m.

**Question 5: The radius and height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the slant height and radius of the cone. (Use π=3.14).**

**Solution:**

Let the ratio of radius and height of a right circular cone be y.

Radius of cone (r) = 5y

Height of cone (h) =12y

**Question 6: The ratio of volumes of two cones is 4 : 5 and the ratio of the radii of their bases is 2 : 3. Find the ratio of their vertical heights.**

**Solution:**

Radius of first cone (r

_{1}) = 2xRadius of second cone (r

_{2}) = 3xVolume of first cone (V

_{1}) = 4yVolume of second cone (V

_{2}) = 5yVolume of a cone = 1/3πr

^{2}h**Question 7: A cylinder and a cone have equal radii of their bases and equal heights. Show that their volumes are in the ratio 3:1.**

**Solution:**

It is given that,

Radius of the cone = Radius of the cylinder = r

Height of the cone = Height of the cylinder = h

(Volume of Cylinder)/(Volume of Cone)=(πr

^{2}h)/(1/3 πr^{2}h)(Volume of Cylinder)/(Volume of Cone)=1/(1/3)

(Volume of Cylinder)/(Volume of Cone)=3/1

Volume of Cylinder : Volume of Cone = 3 : 1.

Hence, ratio of their volumes is 3 : 1.

**Exercise VSAQs.....................**

**Question 1: The height of a cone is 15 cm. If its volume is 500π cm**

^{3}, then find the radius of its base.**Solution:**

It is givens that,

Height of a cone = 15 cm

Volume of cone = 500π cm

^{3}We know, Volume of cone = 1/3 πr

^{2}h500π = 1/3 × π × r

^{2}x 15500 = 1/3 × r

^{2}x 15(500 × 3)/15 = r

^{2}1500/15 = r

^{2}100 = r

^{2}r = √100

r = 10

Hence, the radius of base is 10cm.

**Question 2: If the volume of a right circular cone of height 9 cm is 48π cm**

^{3}, find the diameter of its base.**Solution:**

It is given that,

Height of a cone = 9 cm

Volume of cone = 48π cm

^{3}Volume of cone = 1/3 πr

^{2}h48π = 1/3 × π × r

^{2}× 948 = 1/3 × r

^{2}× 9(48 × 3)/9 = r

^{2}r

^{2}= 16r = √16

r = 4

Radius of base = 4cm

Diameter = 2 Radius = 2 × 4cm = 8 cm.

Hence, the diameter of its base is 8cm.

**Question 3: If the height and slant height of a cone are 21 cm and 28 cm respectively. Find its volume.**

**Solution:**

It is given that,

Height of cone (h) = 21 cm

Slant height of cone (l) = 28 cm

r

^{2}= l^{2}- h^{2}r

^{2}= 28^{2}- 21^{2}r

^{2}= 784 – 441r

^{2}= 343r = 7√7 cm

Volume of cone = 1/3 πr

^{2}hVolume of cone = 1/3 × 22/7 × (7√7)

^{2}× 21Volume of cone = 1/3 × 22 × 49 × 21

Volume of cone = 22 × 49 × 7

Volume of cone = 7546

Hence, the volume of cone is 7546cm

^{3}.**Question 4: The height of a conical vessel is 3.5 cm. If its capacity is 3.3 litres of milk. Find the diameter of its base.**

**Solution:**

It is given that,

Height of a conical vessel = 3.5 cm and

Capacity of conical vessel is 3.3 litres

Change litres in centimetres = 3.3 litres × 1000 = 3300 cm

^{3}Volume of cone = 1/3 πr

^{2}h3300 = 1/3 × 22/7 × r

^{2}× 3.53300 = 1/3 × 22 × r

^{2}× 0.5(3300 × 3)/(22 × 0.5) = r

^{2}900 = r

^{2 }r

^{2}= √900r = 30

Radius of cone is 30cm

Diameter of its base = 2 Radius = 2 × 30 cm = 60cm

Hence, the diameter of conical vessel base is 60cm.